**2. Mathematical Model**

Consider the unsteady three-dimensional MHD non-axisymmetric Homann stagnation point flow of a hybrid Al2O3-Cu/water nanofluid with a stretching/shrinking sheet on the *x*, *y*− plane where *x*, *y* and *z* are Cartesian coordinates with the *z*− axis measured in the horizontal direction and the axes *x* and *y* are in the plane *z* = 0 as illustrated in Figure 1, respectively. We assume that the constant surface temperature *Tw* is stretched and shrunk in the *x* and *y* directions by the velocities *uw* = ε*cx* 1+α*t* and *vw* = ε*cy* 1+α*t* The uniform temperature is given by *T*∞ and *B*0 is introduced to the stretching/shrinking sheet in an orthogonal direction as a transverse uniform magnetic field. Meanwhile, the modified non-asymmetrically free streamflow along the *x*, *y* and *z* axes is described by ([9]):

$$u\_{\varepsilon}(\mathbf{x}) = (a+b)\mathbf{x},\ v\_{\varepsilon}(y) = (a-b)y,\ w\_{\varepsilon}(z) = -2az. \tag{1}$$

**Figure 1.** Flow model and coordinate systems of the physical model.

Here, *a* is the strain rate and *b* is the shear rate of stagnation point flow, correspondingly. By adapting the Tiwari and Das [30] nanofluid model, the continuity, momentum, and the energy equations of the hybrid nanofluid can be written as follows:

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0,\tag{2}$$

$$\frac{\partial \mu}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} = \frac{\partial u\_{\epsilon}}{\partial t} + u\_{\epsilon} \frac{\partial u\_{\epsilon}}{\partial x} + \frac{\mu\_{\text{mf}f}}{\rho\_{\text{mf}f}} \frac{\partial^2 u}{\partial z^2} - \frac{\sigma\_{\text{mf}f}}{\rho\_{\text{mf}f}} B^2 (u - u\_{\epsilon}), \tag{3}$$

$$\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} = \frac{\partial v\_{\varepsilon}}{\partial t} + v\_{\varepsilon} \frac{\partial v\_{\varepsilon}}{\partial x} + \frac{\mu\_{\text{hff}}}{\rho\_{\text{hff}}} \frac{\partial^2 v}{\partial z^2} - \frac{\sigma\_{\text{hnf}}}{\rho\_{\text{hnf}}} B^2 (v - v\_{\varepsilon}), \tag{4}$$

$$\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial \mathbf{x}} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z} = \frac{k\_{\text{lmf}}}{(\rho \mathbb{C}\_p)\_{\text{lmf}}} \frac{\partial^2 T}{\partial z^2} \,. \tag{5}$$

The velocity component in *x*− direction is given by *u*, while *v* is in *y*− direction. Next, the boundary conditions are:

$$\begin{array}{l} t<0: \ u=v=w=0 \text{ for any } \mathbf{x},y,z,\\ t\geq 0: \ u=u\_{\mathbf{w}},v=v\_{\mathbf{w}},w=0, T=T\_{\mathbf{w}} \text{ at } z=0,\\ u\to u\_{\mathbf{c}}=\frac{(a+b)\mathbf{x}}{1+\mathbf{a}t}, v\to v\_{\mathbf{c}}=\frac{(a-b)\mathbf{x}}{1+\mathbf{a}t}, w\_{\mathbf{c}}\to\frac{-2a\mathbf{z}}{1+\mathbf{a}t}, T\to T\_{\mathbf{c}} \text{ as } z\to\infty. \end{array} \tag{6}$$

Note that *T* is the hybrid nanofluid temperature, μ*hn f* is the dynamic viscosity of the hybrid nanofluid, ρ*hn f* is the density of the hybrid nanofluid, *khn f* is the thermal conductivity of the hybrid nanofluid, (ρ*Cp*)*hn f* is the heat capacity of the hybrid nanofluid, σ*hn f* is the electrical conductivity of the hybrid nanofluid, and the time-dependent of a transverse magnetic field is given by *B*<sup>2</sup> = *B*20/(1 + α*<sup>t</sup>*) in detail.

The hybrid nanofluids thermophysical properties are specified in Table 1, as demonstrated by Devi and Devi [41,51]. At this point, φ is the volume fraction of nanoparticles, ρ*f* and ρ*s* are the base fluid density and hybrid nanoparticles, *Cp* is the heat capacity, (ρ*Cp*)*f* and (ρ*Cp*)*s* represent capacitance heating of the base fluid and hybrid nanoparticles, and finally *k f* and *ks* are the thermal conductivities of the base fluid and hybrid nanoparticles, respectively. Meanwhile, the thermophysical properties of the fluid and nanoparticles for aluminum oxide, copper, and the base fluid (water) are given in Table 2.


**Table 1.** Thermophysical properties of hybrid nanofluids (Devi and Devi [41,51]).

**Table 2.** Thermophysical properties of nanoparticles and base fluid (Oztop and Abu-Nada [52]).


Now, pursuing Mahapatra and Sidui [10], the resulting similarity transformation is proposed to achieve the similarity solutions:

$$\begin{cases} u = \frac{\epsilon x^{\prime}(\eta)}{1+\alpha t}, v = \frac{\epsilon y^{\prime}(\eta)}{1+\alpha t}, w = -\sqrt{\frac{\text{vc}}{1+\alpha t}}(f+g), \theta(\eta) = \frac{(T-T\_{\text{ov}})}{(T\_{\text{ov}}-T\_{\text{ov}})}\\ \eta = \sqrt{\frac{\epsilon}{r(1+\alpha t)}}z, \end{cases} \tag{7}$$

where the prime denotes differentiation with respect to η. By substituting (7) into the steady-state Equations (2)–(5), the following ordinary differential equations are obtained:

$$\begin{cases} \frac{\mu\_{\rm mf} / \mu\_f}{\rho\_{\rm mf} / \rho\_f} f''' + A \Big( \frac{1}{2} \eta + f + \mathsf{g} \Big) f'' + A f' - f'^2 - A (\lambda + \mathsf{y}) + (\lambda + \mathsf{y})^2\\ -\mathsf{M} (f' - \lambda - \mathsf{y}) = 0, \end{cases} \tag{8}$$

$$\begin{cases} \frac{\mu\_{\text{hfr}} / \mu\_f}{\rho\_{\text{hfr}} / \rho\_f} \mathbf{g}^{\prime \prime \prime} + A \Big( \frac{1}{2} \eta + f + \mathbf{g} \Big) \mathbf{g}^{\prime \prime} + A \mathbf{g}^{\prime} - \mathbf{g}^{\prime 2} - A (\lambda - \gamma) + (\lambda - \gamma)^2\\ -M(\mathbf{g}^{\prime} - \lambda + \gamma) = 0, \end{cases} \tag{9}$$

$$\frac{1}{\Pr} \frac{k\_{\rm Inf}/k\_f}{\rho \mathbb{C}\_{\rm phnf}/\rho \mathbb{C}\_{pf}} \theta'' + \left( A \frac{1}{2} \eta + f + \mathfrak{g} \right) \theta' = 0,\tag{10}$$

with the boundary conditions (6) which are converted to:

$$\begin{array}{l} f(0) = \mathbf{g}(0) = 0, f'(0) = \mathbf{g}'(0) = \varepsilon, \;\theta(0) = 1, \\\ f'(\eta) \to \lambda + \gamma, \mathbf{g}'(\eta) \to \lambda - \gamma, \theta(\eta) \to 0, \; \text{as } \eta \to \infty. \end{array} \tag{11}$$

In the equations mentioned above, *A* = α/*c* represents the unsteadiness parameter, λ = *a*/*c* is a ratio of the surrounding fluid strain rate to the surface strain rate, γ = *b*/*c* is the surrounding fluid shear rate ratio to the strain rate of the sheet, *M* = <sup>σ</sup>*hn f* /<sup>σ</sup>*f* ρ*hn f* /ρ*f B*20*c* denotes the magnetic parameter and Pr = <sup>ν</sup>*f* /<sup>α</sup>*f* indicates the Prandtl number. The parameter of stretching/shrinking is meant by ε where ε > 0 determines the stretching sheet, while ε < 0 reflects the shrinking sheet. The related quantities of interest in this study are the skin friction coefficient, *Cf x* and *Cf y* along the *x*− and *y*− directions and the local Nusselt number *Nux*, which is specified as

$$\mathbf{C}\_{f\mathbf{x}} = \frac{\mathbf{\tau}\_{\text{uv}\mathbf{x}}}{\rho\_f \mathbf{u}\_\varepsilon^{2}}, \; \mathbf{C}\_{fy} = \frac{\mathbf{\tau}\_{\text{uv}y}}{\rho\_f \mathbf{v}\_\varepsilon^{2}}, \; \mathbf{N} \mathbf{u}\_\mathbf{x} = \frac{\mathbf{x}q\_w}{k\_f(T\_w - T\_\infty)},\tag{12}$$

where τ*wx*, <sup>τ</sup>*wy* are the shear stresses along the *x*<sup>−</sup>, *y*− axes and *qw* represents the heat flux, correspondingly. Such terms can be defined by

$$
\pi\_{\rm uvx} = \mu\_{\rm lmf} \left( \frac{\partial u}{\partial z} \right)\_{z=0}, \tau\_{\rm wy} = \mu\_{\rm lmf} \left( \frac{\partial v}{\partial z} \right)\_{z=0}, \\
q\_{\rm w} = -k\_{\rm lmf} \left( \frac{\partial T}{\partial z} \right)\_{z=0}. \tag{13}
$$

By prompting Equations (7), (12) and (13), we get:

$$\begin{cases} \left(\lambda + \gamma\right)^{3/2} \text{Re}\_x^{1/2} \mathbb{C}\_{f\_x} = \frac{\mu\_{\text{hf}}}{\mu\_f} f''(0), (\lambda - \gamma)^{3/2} \text{Re}\_y^{1/2} \mathbb{C}\_{f\_y} = \frac{\mu\_{\text{hf}}}{\mu\_f} \text{g}''(0),\\ \left(\lambda + \gamma\right)^{1/2} \text{Re}\_x^{-1/2} \text{Nu}\_x = -\frac{k\_{\text{hf}}}{k\_f} \theta'(0), \end{cases} \tag{14}$$

where Re*x* = (*a*+*b*)*x*<sup>2</sup> (<sup>1</sup>+α*<sup>t</sup>*)<sup>ν</sup>*f* and Re*y* = (*a*+*b*)*y*<sup>2</sup> (<sup>1</sup>+α*<sup>t</sup>*)<sup>ν</sup>*f* are the local Reynolds number along the *x*− and *y*− directions, respectively.
